In this assignment you will model the orbital debris collision hazard for spacecraft in Earth orbit. This will require you to program formulas representing the flux (impacts/ unit time/ unit area) of debris, numerically integrate these formulas forward in time, and perform an analysis where different inputs will affect the results (a parametric study).

Background

The flux of orbital debris on a spacecraft can be represented by a fairly simple formula (the boxed equation below) that depends on two sets of information: Information about the spacecraft such as altitude, inclination, and orientation; and, information about the environment such as particle sizes, growth rates, and the effects of the solar cycle (through changing atmospheric drag).

For a little perspective realize that while a .1-mm particle may cause serious surface erosion, a 1-mm particle can be very damaging. A 3-mm particle traveling at 10 km/s has the kinetic energy of a bowling ball thrown at 100 km/hr and a 1-cm particle would have the energy of a 180-kg safe thrown at the same speed. The U.S. space shuttles have already changed their orbits several times to avoid large debris. There has been pitting of tiles, and the loss of several panes of its multi-paned windshield due to impact with a small fleck of paint.

The natural meteoroid environment has historically been a design consideration for spacecraft. Meteoroids are part of the interplanetary environment and sweep through Earth Orbital space at an average speed of 20 km/s. At any one instant, a total of 200 kg of meteoroid mass is within 2000 km of the Earth's surface. Most of this mass is concentrated in 0.1-mm meteoroids.

Within this same 2000 km above the Earth's surface, however, are an estimated 3,000,000 kg of man-made orbiting objects. These objects are mostly in high inclination orbits and sweep past one another at an average speed of 10 km/s. Most of the mass is concentrated in approximately 3000 spent rocket stages, inactive payloads and a few active payloads. A smaller amount of mass, approximately 40,000 kg, is in the remaining 4000 objects currently being tracked by U.S. Space Command radars. Most of the objects are the result of more than 90 on-orbit satellite fragmentation. Recent ground telescope measurements of orbital debris combined with an analysis of hypervelocity impact pits on the returned surfaces of the Solar Maximum Mission (SMM) satellite indicate a total mass of approximately 2000 kg for orbital debris of 1 cm or smaller and approximately 300 kg for orbital debris smaller than 1 mm. This distribution of mass and relative velocity is sufficient to cause the orbital debris environment to be more hazardous than the meteoroid environment to most spacecraft orbiting below 2000 km altitude.

While low-altitude debris will fall to the Earth due to atmospheric drag, it is quickly replenished by particles higher up and from collisions. It has been proposed that if too much debris gets into orbit, collisions could cause an increasing number of breakups, leading to an exponential growth in the number of particles. Such a catastrophic chain-reaction has been referred to as the "Kessler Syndrome". Even with the envisioned growth in launch rate, the growth of orbital debris can be greatly reduced by de-orbiting spent rocket stages and satellite at the end of their useful lifetimes. Better design and management can also reduce the likelihood of explosions (often related to propulsion systems) and reduce the amount of debris likely to be generated in a collision. Orbiting robots have also been proposed to scavenge for old satellites so they could be recycled before they disintegrate, perhaps to be used as small source of materials in future space development.

1. Prepare a multi-curve plot that illustrates the average number of impacts on a spacecraft (y-axis) vs. altitude in km (x-axis) for particle sizes of 0.1, 0.2, and 0.3 cm. The range of heights should be 100 km to 2,000 km, and assume the 10-year time interval: t1=1985, t2=1995. Use the default values listed with the equations, i.e., k=1; random tumbling surface; I = 28.5 degrees inclination, etc.

2. Make at least two other graphs of your choice, in each one varying the effect of altering one variable - change k, S, t2, i () or p. For comparison keep d = {0.1, 0.2, 0.3 cm} or d = 0.1 cm depending on your graph. Explain your choices.